What would happen to an agent trained using Markov Decision Process if the goal node changes?
It is possible, at design time for a reinforcement learning problem, to allow for changes within an environment. You can make any element into a variable property of the state, that the agent can realistically be told at the start or sense from the environment.
If you do add new variable to model the possibility of change:

It allows the agent to learn to solve a more general problem where the chosen property can vary.

It increases the size of the state space.

It requires training to include variations of the new variable.


Usually this also increases the time taken to train.
It is not always possible to use a state variable for the task - perhaps a goal state is effectively hidden from the agent and the purpose of training is for it to be discovered. In which case, you will require at least some re-training. It may be faster to start with the existing trained agent if the difference is not large.
If you cannot simply extend the state representation, and the environment changes in a small enough way, then it may also be possible to use an agent which continuously explores which will re-train itself over time in repsonse to changes in the environment. The DynaQ+ algorithm is an example of a method which is designed to explore and find changes in the agent's environment to allow for this kind of online retraining when things change.